**Spin-lattice relaxation via quantum tunneling in diluted crystals of Fe**

**4**

**single-molecule magnets**

A. Repoll´es,1,2_{A. Cornia,}3,*_{and F. Luis}1,2,*†*

1* _{Instituto de Ciencia de Materiales de Arag´on, CSIC-Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain}*
2

*3*

_{Departamento de F´ısica de la Materia Condensada, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain}

_{Dipartimento di Scienze Chimiche e Geologiche and UdR INSTM, Universit`a di Modena e Reggio Emilia, via G. Campi 183,}*41125 Modena, Italy*

(Received 3 December 2013; published 26 February 2014)

We investigate the dynamic susceptibility of Fe4 *single-molecule magnets with integer spin (S*= 5) in the

form of pure crystals as well as diluted in crystals of isostructural, but nonmagnetic, Ga4 clusters. Below

approximately 1 K, the spin-lattice relaxation becomes dominated by a temperature-independent process. The
*spin-lattice relaxation time τ measured in this “quantum regime” is 12 orders of magnitude shorter than the*
characteristic time scale of direct phonon-induced processes but agrees with the relaxation times of pure (i.e.,
not assisted by phonons) spin tunneling events. The present results show that the latter phenomenon, despite
conserving the energy of the ensemble of electronic and nuclear spins, drives the thermalization of electronic spins
at very low temperatures. The spin-lattice relaxation time scales with the concentration of Fe4, thus suggesting

that the main effect of dipolar interactions is to block tunneling. The data show therefore no evidence for the
contribution of collective phonon emission processes, such as phonon superradiance, to the spin-lattice relaxation.
DOI:10.1103/PhysRevB.89.054429 *PACS number(s): 75.45.+j, 76.30.Kg, 75.50.Xx, 75.40.Gb*

Single molecule magnets (SMMs) [1] are high-spin mag-netic molecules comprising one or more metal centers encap-sulated in a shell of organic ligands. They provide a very attractive workbench for research on quantum phenomena in magnetism, such as magnetization tunneling [2–4], Berry phase interferences [5], quantum spin coherence [6–8], and quantum phase transitions [9]. Although the underlying physics governing such phenomena is fairly well understood, some fundamental questions still remain open.

A particularly intriguing puzzle concerns the nature of
spin-lattice relaxation (SLR) mechanisms that bring spins to
thermal equilibrium at very low temperatures, typically for
*T* 1 K, when thermally activated processes [10–12] die
out. Under these conditions and near zero magnetic field,
spins predominantly flip by tunneling across the anisotropy
energy barrier. Hyperfine interactions with environmental
nuclear spins (e.g., those of the metal centers themselves
and of other atoms present in the outer ligand shell) can
compensate for the magnetic bias associated with intercluster
dipolar interactions, thus bringing some molecular spins close
to resonance conditions and enabling them to tunnel [13–15].
Tunneling modifies the magnetization but conserves the energy
of the ensemble of nuclear and electronic spins. By contrast,
SLR requires that magnetic energy is either released to or
absorbed from the lattice, e.g., via the direct emission or
absorption of a phonon [16]. Since the latter events can be
extremely slow at low magnetic fields, it can be expected that
magnetization dynamics and SLR take place at very different
time scales [17].

Yet, experiments performed on different SMMs [18–20]
give SLR times that are close to the expected tunneling times,
thus suggesting that the thermalization of electronic spins is
*dictated by tunneling fluctuations. A plausible, yet qualitative*
interpretation of the existing experimental evidences is that

*_{acornia@unimore.it}
*†*_{fluis@unizar.es}

SLR takes place via phonon superradiance [21] from
partic-ular spin configurations, which the spin ensemble “visits”
via tunneling processes [22]. This phenomenon has been
investigated on lanthanide ions diluted in diamagnetic crystals
[22,23]. Unfortunately, the results are obscured by either the
*dependence of the quantum tunnel splitting on concentration*
(for Kramer’s ions) or the existence of large hyperfine
*splittings, which dominate the physics at very low T . These*
effects prevent any simple, quantitative comparison of SLR
experiments with theoretical predictions for spin tunneling.

*Crystals of polynuclear SMMs can provide a valuable*
alternative. However, synthesizing crystalline solid solutions
of intact polynuclear species and their diamagnetic analogues
is a very challenging task. First, preparing isostructural but
diamagnetic variants of known SMMs may be difficult or
impossible, especially for mixed-valent species. Second, the
solid solution must crystallize without metal scrambling, i.e.,
without any exchange of metals that produces mixed-metal
species. The first successful synthesis of diluted polynuclear
SMMs in crystalline form was achieved [24] with tetra-iron
molecular clusters (see Fig.1), which are known to be highly
stable and robust [25–27]. A fraction of Fe4 *clusters (S*= 5)

is replaced with nonmagnetic but structurally equivalent Ga4 clusters, thereby providing a means of controlling the

characteristic energy scale of dipolar couplings. In addition, the vast majority (98%) of Fe atoms carry no nuclear spin. These crystals are therefore model systems to study the relationship between SLR and tunneling, and to elucidate the role played by dipolar interactions. In this paper, we study the SLR times of pure as well as diluted Fe4 crystals and

compare the results with existing theories for SLR and spin tunneling.

Single-crystalline samples of [(Fe4)x(Ga4)1−x(L)2(dpm)6]·

C6H6, hereafter referred to as (Fe4)x(Ga4)1−x *with x= 1.00*

*and 0.05, were prepared as described in Ref. [*24].
Com-pounds of the series (Fe4)x(Ga4)1*−x* are isomorphous and

*crystallize in the monoclinic space group C2/c with four*
equivalent molecules per unit cell. Only minor variations

(a)

(c)

(b)

FIG. 1. (Color online) (a) Molecular structure of Fe4 viewed

along the normal to the metal plane, which is taken to coincide with
*the easy magnetic axis (Z). (b) Magnetic energy level scheme of*
Fe4*(dotted horizontal lines). Levels are labeled by the m quantum*

*number associated with the Z component of the spin in the absence of*
tunneling effects. The solid line is the classical double-well potential
*as a function of m. (c) Crystal structure of Fe*4 SMMs diluted in a

diamagnetic crystal of Ga4 clusters. Color code: Fe, light orange;

Ga, dark blue; O, red; C, gray. Hydrogen atoms and benzene lattice
molecules have been omitted for clarity. The normal to the metal
*plane (Z) forms an angle of 1.44*◦*with a*∗.

of unit cell parameters are observed in the series, with
*a 1% contraction of unit cell volume from x= 1.00 to*
*x* = 0 at 120 K. Magnetic measurements in the region of
*1.8 K* * T 300 K were performed on powder specimens*
using a commercial SQUID magnetometer. Ac susceptibility
measurements have been extended down to 13 mK using a
*μ-SQUID susceptometer installed inside the mixing chamber*
of a 3He-4He dilution refrigerator [28,29]. In these
experi-ments,*∼800 × 400 × 200 μm*3single crystals were directly
*placed on top of one of the two μ-SQUID loops. The easy*
*magnetic axis made an angle ψ* 56◦ with respect to the ac
*excitation magnetic field (h*ac*<*1 mOe). The frequency of the

*latter was varied between ω/2π* *= 0.01 Hz and 2 × 10*5_{Hz.}

Representative examples of the in-phase *χ* and
*out-of-phase χ* magnetic susceptibilities of Fe4 and

(Fe4)*0.05*(Ga4)*0.95*, measured as a function of frequency at

fixed temperatures, are shown in Fig. 2. They show the typical behavior of a SMM, with a well defined transition from equilibrium conditions, at sufficiently low frequencies, to adiabatic conditions, in the opposite frequency limit. The

FIG. 2. (Color online) Frequency dependent susceptibility
isotherms of pure Fe4 (left) and of (Fe4)*0.05*(Ga4)*0.95* (right) at

several temperatures. Solid lines are least-square Cole-Cole fits [cf.
Eqs. (1) and (2)]. The insets show the temperature dependence of
*the reciprocal in-phase susceptibility jump 1/χ . Solid lines are*
least-square Curie-Weiss fits, with the parameters shown in each
graph.

*transition takes place approximately when ωτ 1, where τ*
is the SLR relaxation time, and coincides with the maximum
*of χ*. We find (see the insets of Fig. 2) that the in-phase
*susceptibility “jump” χ (i.e., the net variation between*
its high and low frequency limits) follows Curie-Weiss law
*χ C/(T − θ), where C is the Curie constant and θ is the*
Weiss temperature that depends on the average strength of
intermolecular magnetic interactions. This shows that linear
*susceptibility experiments measure SLR to thermal *
*equi-librium and not spin-spin relaxation within the “spin-bath.”*
This experimental situation contrasts sharply with that met
in spin tunneling experiments, be it magnetization hysteresis
or Landau-Zener relaxation measurements. In the latter case,
magnetization jumps that occur at tunneling resonances (level
crossings) link two nonequilibrium spin configurations. As
*expected, C scales with x since it is proportional to the number*
*of spins per unit of sample mass. In addition, θ of Fe*4is about

six times larger than that of (Fe4)*0.05*(Ga4)*0.95*, thus showing

that interactions become significantly reduced by dilution.
*Above T* 1 K, SLR times were obtained by fitting
susceptibility isotherms with Cole-Cole functions [30]

*χ**= χS*+*(χT* *− χS)*
1*+ (ωτ)βcos (βπ/2)*
1*+ 2 (ωτ)βcos (βπ/2)+ (ωτ)2β* (1)
*χ*= *(χT* *− χS) (ωτ )*
*β*
*sin (βπ/2)*
1*+ 2 (ωτ)βcos (βπ/2)+ (ωτ)2β,* (2)
*where χT* *and χS*are the equilibrium and adiabatic
*suscepti-bilities, respectively, and β gives information on the width of*
*the distribution of relaxation times. We find that β ranges from*
*0.85 at T* *= 2.1 K to 0.79 at T = 1.2 K for pure Fe*4and from

*0.82 at T* *= 2.1 K to 0.72 at T = 1.2 K for (Fe*4)*0.05*(Ga4)*0.95*.

*For temperatures below 1 K, the maxima of χ* occur
at frequencies lower than our minimum experimental limit

*(0.01 Hz). Even so, SLR times can be approximately estimated*
*using the relation τ* * [r/(sin βπ/2 − r cos βπ/2)](1/β)/ω,*
which follows from Eqs. (1) and (2*). Here, r* *= χ**/(χ**− χS),*
*and β and χS*have been fixed to their respective values found
*at T* *= 1.2 K, as they are expected to vary only weakly with T .*
The results are shown in Fig. 3. SLR relaxation times of
both samples follow a thermally activated behavior above
*approximately 1.2 K, where τ* * τ*0*exp(U/k*B*T*), gradually

approaching a nearly constant value below 1 K. However, some
differences are seen between the two samples. The activation
*energy U is largest for the pure Fe*4crystal, which also has the

*shortest prefactor τ*0. The same trend was found by in-field

*(0.1 T) ac measurements on powder samples [*24]. In the
temperature-independent, or quantum, regime, the relaxation
of the pure Fe4 compound is about 10 times slower than that

of the diluted sample.

We next compare these results with predictions for the characteristic time scales of phonon-induced relaxation and pure tunneling processes. Let us first consider the spin Hamiltonian of an isolated Fe4cluster. Magnetic interactions

between the four Fe3+ions within the cluster core, shown in
Fig.1*, give rise to a ground state with a net spin S*= 5 and a
*gyromagnetic ratio g= 2.005(4). Interactions with the crystal*
field and with magnetic fields split this multiplet. These effects
can be described by the following spin Hamiltonian:

*H = B*0

2*O*20*+ B*22*O*22*+ B*40*O*40*− gμ*B−→*S* ·−→*H* *+ H**,* (3)

*where B*_{2}0*/k*B*= 0.200(2) K, B*22*/k*B*= 0.023(1) K, and*

*B*0

4*/k*B= 9(4) × 10−6K are magnetic anisotropy parameters,

FIG. 3. (Color online) Spin-lattice relaxation times of pure Fe4

(top) and (Fe4)*0.05*(Ga4)*0.95*(bottom). Solid symbols were obtained

from Cole-Cole fits [Eqs. (1) and (2)] of isothermal susceptibility vs frequency measurements. Open symbols were obtained from the ratio

*χ**/(χ**− χS*), as described in the text. Solid lines are phonon-induced

relaxation times calculated by solving a Pauli master equation for the population of magnetic energy levels. The dotted lines are spin tunneling times predicted by Eq. (4).

determined from the fit of spectroscopic data, and *H* is an
effective term arising from interactions that mix states from
different multiplets [24]. The ensuing energy level scheme
is shown in Fig.1. The population of the ground state level
*doublet, associated with the maximum projections m= ±S of*
*the spin along the anisotropy axis Z, becomes larger than 99%*
*for T* 1 K. Under these conditions, each Fe4cluster behaves

effectively as a two level system, with an energy splitting
*E= (*2*+ ξ*2)*1/2, where is the quantum tunnel splitting*
induced by off-diagonal terms in Eq. (3*) and ξ 2gμ*B*SHz*is

the energy bias associated with longitudinal magnetic fields.
In Ref. [24*], it was reported that /k*B 9 × 10−7K and that

it depends only weakly on transverse magnetic fields.
In order to realistically account for the influence of
intermolecular magnetic interactions, we have also evaluated
the distributions of internal dipolar fields in concentrated and
diluted samples. Two model crystal samples were tested: (i)
a spherical portion of crystal lattice with radius 300 ˚A and
(ii) a model crystal mimicking the typical crystal shape bound
by faces (001), (111), and (111) (and their equivalents) at
distances of 20, 16, and 8 interplanar spacings, respectively,
from the origin. The two models comprised comparable
num-bers of complete molecules (*∼4.3 − 4.5 × 10*4_{). To describe}

magnetically-diluted samples, a fraction 1*− x of molecules*
was randomly chosen and removed from the ensemble.
Molecular spins were randomly assigned to be either in the
*m= +5 or in the m = −5 states, with equal probability.*
The dipolar field −*H*→d*= (Hd,X,Hd,Y,Hd,Z*) acting on each

molecule was computed from the dipolar fields experienced
*by its constituent ions. The longitudinal component Hd,Z*

was evaluated as the weighted average over the four ions,
*whereas for the transverse (XY ) component an unweighted*
average was used [24,31]. Owing to the zero magnetization,
the demagnetizing field vanishes, and calculations made on
the spherical and nonspherical model samples give virtually
identical results (see Fig.4). For the pure Fe4sample, the bias

field distribution is Gaussian with FWHM of approximately
20 mT, while for the diluted sample it is non-Gaussian
and much narrower (FWHM∼ 2 mT). The transverse fields
*distribution for x*= 1 is rather broad and extends up to 30 mT
with a peak at ca. 10 mT. By contrast, the diluted sample
features a more structured distribution extending up to about
*10 mT, with a main peak at 0.5 mT and secondary maxima at*
higher field values. These arise from pairs of neighboring Fe4

complexes in the lattice.

SLR times have been numerically computed by applying
a Pauli master equation to calculate the time-dependent
populations of the magnetic energy levels of Eq. (3) and,
from them, the frequency-dependent susceptibility [10,32].
*The spin-phonon interaction Hamiltonian was 3B*20*{(xz*+

*ωxz)⊗ [SxSz+ SzSx*]*+(yz+ ωyz)⊗ [SySz+ SzSy]}, where*
*iz* *and ωiz* represent phonon-induced strains and rotations,
respectively. The overall scale of all relaxation rates is fixed
*by a constant parameter q∝ n*ph*/c*5s*, where n*phis the number of

*relevant low-energy phonon modes and c*sis the speed of sound

that describes, in the simplest manner, the dispersion relation
*ω= c*s*k*of phonon modes. This parameter was determined by

*fitting τ measured at T* = 2 K on pure Fe4. The speed of sound

(a)

(b)

FIG. 4. (Color online) (a) Shaded areas, distribution of dipolar
*bias fields Hd,Z* calculated for magnetically unpolarized crystals

of Fe4 (light red) and (Fe4)*0.05*(Ga4)*0.95* (dark blue); lines,

*least-squares fits of these distributions using either a Gaussian (x*= 1)
*or a Lorentzian (x= 0.05) function. (b) Distributions of transverse*
dipolar magnetic fields calculated for the same model samples.

*n*_{ph} *= 3 acoustic modes) to 1.5 × 10*5_{cm/s (if all 768 acoustic}

and optical modes contribute). These values are comparable to
those previously found for other SMMs [18,20,32]. In order to
“tune” the ground state tunnel splitting to its actual value while
keeping the numerical calculations tractable, we simulated
the effect of *H* *by introducing an off-diagonal B*_{6}5*O*_{6}5 term
*into the giant spin Hamiltonian of the S*= 5 multiplet, with
*B*_{6}5*/k*B*= 1.05 × 10*−6K. The susceptibility has been averaged

*over the bias field distribution P (Hd,Z*) shown in Fig.4. For

simplicity, transverse dipolar interactions have been taken into
account by introducing an average transverse magnetic field
*H _{d,⊥}(1/*√

*2,1/*√

*2,0), with Hd,⊥= 10 mT and 0.5 mT for the*

concentrated and diluted samples, respectively.

The results of these calculations are shown in Fig.3. Above
1 K, they account well for the experimental data: At any
temperature, thermally activated relaxation becomes faster
*as the magnetic concentration x decreases. In particular, the*
*effective activation energies (obtained from fits for T > 1 K)*
*are U/k*B* 12.9 K for x = 1 and U/k*B* 12 K for x =*

*0.05. Both values are smaller than the “classical” activation*
*energy U*cl*/k*B= 15.0 K extracted from spectroscopic data

through Eq. (3) [24], thus showing that tunneling via thermally
*activated spin states strongly influences τ in this regime. By*
contrast, the same model completely fails to account for the
SLR observed below 1 K. It predicts a monotonic increase
*of τ with decreasing temperature down to approximately*
*0.1 K, where it tends to saturate to an astronomically*
long (1013 _{s) value, associated with direct phonon-induced}

processes. Clearly, a different process drives SLR below 1 K. At very low temperatures, spin dynamics is fully dominated by pure quantum spin tunneling events. According to the Prokof’ev and Stamp model [13], the average tunneling rate is approximately given by

=

2

*P(ξ*d*),* (4)

*where P (ξ*d)*= P (Hd,Z)/(2gμ*B*S) is the distribution of dipolar*

*energy bias. Tunneling times *−1 obtained from Eq. (4)
for pure and diluted samples agree very well with the SLR
times measured below 1 K. In particular, at zero applied
field, Eq. (4) predicts (dotted lines in Fig. 3*) that *−1
*approximately scales with the width of P (ξ*d*), i.e., with x,*

as the SLR time indeed does. This result contrasts with the
*decrease of τ with increasing Er*3+ concentration observed
in crystals of Na9[ErxY1*−x*(W5O18)2]·yH2O [22]. Yet, the

markedly different behavior of these materials can also be
reconciled with the above interpretation, on the basis of
Eq. (4*). While of Fe*4 is approximately independent of

*H _{d,⊥}, of Er*3+

*, a Kramer’s ion with J*

*= 15/2, vanishes*

*unless the local Hd,*⊥

*= 0, thus it is expected to increase with*

*concentration. Also, the slight temperature dependence of τ in*
the quantum regime can be associated with gradual changes in
the distribution of dipolar bias.

The thermalization of spins plays a crucial role in funda-mental phenomena, such as the attainment of magnetically ordered states, as well as in their application as magnetic refrigerants or thermometers, thus its relevance can hardly be overestimated. Our experiments show that spin-lattice relax-ation of anisotropic spins takes place, at very low temperatures, at rates that quantitatively match predictions for spin tunneling processes, thus much faster than those of direct spin-phonon processes. However, the precise mechanism by which the spins that flip by tunneling exchange energy with the lattice remains obscure. For a fixed tunneling rate, such as that of Fe4, dipolar

interactions mainly slow down the relaxation process by taking most spins off-resonance. Therefore, experiments provide no evidence supporting the contribution of collective emission of phonons to spin-lattice relaxation. The last piece of the puzzle thus remains to be found.

This paper has been partly funded by the Spanish MINECO
(Grant No. MAT2009-13977-C03), the Gobierno de Arag´on
(project MOLCHIP), the European Union (ERANET project
*NanoSci-ERA: Nanoscience in European Research Area*
SMMTRANS), the Italian MIUR (PRIN2008 project) and
the Universit´e Joseph Fourier (Grenoble, France) through a
visiting professorship to A.C.

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